I mean... What do "new users" need to see? Who is a new user? Why are they here? Why are they on the home page? Why would what they want to see differ heavily from what experienced users want to see?
I don't have answers to any of these questions... As an experienced user I can suggest that new users aren't properly informed what the site is and what role they are expected to play on it, but would explaining that on the home page make sense? Maybe if they haven't ticked a box yet? Maybe instead what would be most important is a list of their own questions and answers, whether they're in SG or live. Maybe it should show newly approved posts from SG so that they're only presented posts that have been reviewed as being of at least decent quality so they don't get the wrong impressions of what the site is.
But ultimately... I don't know. I'm not a new user, ;) None of these would be a good experience for me.
Is there anything in particular we should add that is the right action for new users to take?
Prominently displaying the "What is on topic?" copy from the help center somewhere would probably be a good idea, assuming it isn't still the default like it is on some super tiny sites.
Alternatively, is there something we should take away that you find is not helpful?
I don't think reputation, badges, or tags are relevant here. As gamification elements they should definitely exist to some extent, but maybe at the very top of the homepage isn't that. I'd prefer the gamification aspect come online once they've created useful content, rather than it being what drives them to begin.
Also, if you had to pick between redesigning the post summary or work done on the tour page, which would you prefer?
The post summary is likely to have the most impact given it's the one everyone who is asking their first question is guaranteed to reach.